BeeTheory · Foundations · Technical Note I
A Regularized Wave Function for BeeTheory
A minimal, single-parameter refinement of the BeeTheory wave function that removes the singularity at the origin while preserving every prediction of the theory at larger scales. This note establishes the mathematical foundation needed to extend BeeTheory rigorously from elementary particles to galaxies.
The BeeTheory wave function
$$\psi(r) = \frac{1}{N}\,\exp\!\left(-\frac{\sqrt{r^2 + a^2}}{a}\right)$$
where $a$ is the natural length scale of the particle
(for hydrogen: $a = a_0 = 5.29 \times 10^{-11}$ m, the Bohr radius)
This formula carries three properties that make BeeTheory a complete and well-defined theory at every scale, from the subatomic to the galactic:
| Property | Value at $r = 0$ | Behavior for $r \gg a$ |
|---|---|---|
| Wave function $\psi(r)$ | $e^{-1} \approx 0.368$ (finite) | $\to e^{-r/a}$ (matches the original BeeTheory postulate) |
| Laplacian $\nabla^2\psi$ | $-3\,e^{-1}/a^2$ (finite) | $\to e^{-r/a}/a^2$ (asymptotically identical) |
| Free parameters | One ($a$ alone) | No additional length scale |
1. Why regularize?
BeeTheory, in its original formulation (Dutertre 2023), postulates that every elementary particle is described by a radial exponential wave function:
Original BeeTheory postulate
$$\psi_0(r) = N_0 \cdot e^{-r/a}$$
This form is elegant and mathematically transparent, and it correctly captures the long-range behavior of the wave field. However, when expressed in spherical coordinates and acted upon by the Laplacian operator that appears in the Schrödinger equation, an artifact emerges at the origin:
Laplacian of the original form
$$\nabla^2 \psi_0(r) = \psi_0(r) \cdot \left(\frac{1}{a^2} – \frac{2}{r\,a}\right)$$
The term $-2/(r\,a)$ grows without bound as $r \to 0$. This is a familiar feature of point-like idealizations in physics — the same kind of singularity that appears in the Coulomb potential, and one that is routinely handled in nuclear and atomic physics through regularization techniques. The regularized BeeTheory wave function described below applies precisely this kind of established technique.
2. The regularization principle
The principle is elegantly simple: replace $r$ with $\sqrt{r^2 + a^2}$ inside the exponential. This substitution is a classical regularization technique used throughout theoretical physics — notably for softened Yukawa potentials in particle physics and pseudopotentials in quantum chemistry. It introduces no new physical scale: the regularization length is the particle’s own characteristic length $a$.
The substitution
$$r \longrightarrow \sqrt{r^2 + a^2}$$
The physical interpretation is natural and consistent with BeeTheory’s foundational view of particles as extended wave structures: a particle whose characteristic size is $a$ cannot have a feature smaller than $a$ itself. The wave field at the core of the particle is smooth at the scale of its own coherence length. This is a strengthening of the original postulate, not a departure from it.
Behavior at both limits
Near the origin ($r \ll a$): using $\sqrt{r^2 + a^2} \approx a + r^2/(2a)$, we obtain
$$\psi(r) \approx e^{-1} \cdot e^{-r^2/(2a^2)}$$
The wave function smoothly transitions to a Gaussian near the center, with finite value $e^{-1}$ at $r = 0$. The probability density is well-defined throughout the entire interior of the particle.
Far from the origin ($r \gg a$): using $\sqrt{r^2 + a^2} = r\sqrt{1 + (a/r)^2} \approx r + a^2/(2r)$, we obtain
$$\psi(r) \approx e^{-r/a} \cdot e^{-a/(2r)} \;\longrightarrow\; e^{-r/a}$$
We recover exactly the exponential decay of the original BeeTheory postulate. Every prediction of BeeTheory at distances larger than the particle’s own scale — and that includes every atomic, planetary, and astrophysical application of the theory — is preserved without modification.
3. Numerical verification
The table below compares the original wave function $\psi_0$ and the regularized $\psi$, together with their Laplacians, at various distances expressed in units of $r/a$:
| $r/a$ | $\psi_0$ (original) | $\nabla^2\psi_0$ | $\psi$ (regularized) | $\nabla^2\psi$ |
|---|---|---|---|---|
| 0.001 | 0.999 | −1997 | 0.368 | −1.104 |
| 0.01 | 0.990 | −197.0 | 0.368 | −1.103 |
| 0.1 | 0.905 | −17.19 | 0.366 | −1.085 |
| 0.5 | 0.607 | −1.820 | 0.327 | −0.753 |
| 1.0 | 0.368 | −0.368 | 0.243 | −0.308 |
| 2.0 | 0.135 | 0.000 | 0.107 | −0.020 |
| 5.0 | 0.0067 | 0.004 | 0.0061 | 0.003 |
| 10.0 | 4.5×10⁻⁵ | ≈ 0 | 4.3×10⁻⁵ | ≈ 0 |
The regularized Laplacian remains finite everywhere, with magnitude of order $1/a^2$ near the origin, and converges to the original beyond $r \approx 5a$. The refinement is strictly local: confined to a neighborhood of the particle of size $\sim a$, and entirely invisible at every larger scale.
4. The analytical Laplacian
The derivation is direct. Setting $s(r) = \sqrt{r^2 + a^2}$ and $\psi(r) = N^{-1}\,e^{-s/a}$, the radial derivatives are:
Derivatives of s(r)
$$s'(r) = \frac{r}{s}, \qquad s”(r) = \frac{a^2}{s^3}$$
Applying the chain rule and the Laplacian in spherical coordinates $\nabla^2 = \partial_r^2 + (2/r)\partial_r$ for a radially symmetric function, we obtain the compact closed form:
Laplacian of the BeeTheory wave function
$$\boxed{\;\nabla^2 \psi(r) = \psi(r) \cdot \left[\frac{r^2}{a^2 s^2} – \frac{3}{a\,s} + \frac{r^2}{a\,s^3}\right]\;}$$
This expression is finite everywhere, including at $r = 0$. Evaluation at the two natural limits:
| Limit | $s(r)$ | $\nabla^2 \psi(r)$ |
|---|---|---|
| $r \to 0$ | $s \to a$ | $\psi(0) \cdot (-3/a^2) = -3\,e^{-1}/a^2$ |
| $r \to \infty$ | $s \to r$ | $\psi(r) \cdot (1/a^2 – 3/(r\,a))$ |
At large distance, the Laplacian recovers the form of the original BeeTheory expression $\psi \cdot (1/a^2 – 2/(r\,a))$ up to a $1/r$ correction that vanishes rapidly. The difference is negligible beyond $r$ greater than $5a$ — far inside any physical regime relevant to gravitational or astrophysical applications.
5. What this unlocks for BeeTheory
A theory now well-defined at every scale
BeeTheory’s Schrödinger equation, applied to the regularized $\psi$, has finite kinetic energy $-\frac{\hbar^2}{2m}\nabla^2\psi$ at every point in space. The wave-based mechanism of gravity is now mathematically rigorous from the interior of a single particle to the largest galactic scales. This is the technical foundation that bridges the atomic and the cosmic in a single, consistent framework.
All long-range predictions preserved
The asymptotic behavior of $\psi$ is identical to the original BeeTheory wave function. Every prediction at length scales larger than the atomic radius is preserved without modification — including the inverse-square gravitational law derived from the spherical Laplacian, the shell theorem allowing macroscopic bodies to be treated as point particles, and the extension to extended distributions of matter at galactic scales. The refinement strengthens the foundation without disturbing the structure built upon it.
What comes next
With the wave function now rigorously defined everywhere, the central derivation of BeeTheory — the application of the Schrödinger equation to a pair of interacting waves yielding the gravitational $1/R$ potential — can be reformulated in full mathematical rigor, with every step explicit and every coefficient determined from first principles. This is the subject of the next technical note in this series.
6. Summary in three lines
1. The BeeTheory wave function is $\psi(r) = N^{-1}\,\exp\!\left(-\sqrt{r^2+a^2}/a\right)$.
2. Its Laplacian is finite everywhere, taking the value $-3\,e^{-1}/a^2$ at the origin.
3. Beyond $r \approx 5a$, it is numerically indistinguishable from the original $e^{-r/a}$.
References. Dutertre, X. — Bee Theory™: Wave-Based Modeling of Gravity, v2, BeeTheory.com (2023). Original postulate. · Schwabl, F. — Quantum Mechanics, 4th ed., Springer (2007). Regularization of singular potentials. · Hellmann, H. — A New Approximation Method in the Problem of Many Electrons, J. Chem. Phys. 3, 61 (1935). Historical origin of regularized pseudopotentials in quantum mechanics.
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